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Math Homework Help

Two Similar Cylinders

published on March 5th, 2008 . by Vanaja


The two cylindrical pans are similar. The diameter of the smaller pan is equal to the radius of the larger pan. How many of these smaller cans could fill the larger can?

similar-cylinders.JPG

Hint: Since the two cylinders are similar, their dimensions are in the same ratio. It is given that the diameter of the smaller pan is same as the radius of the larger pan. That is the radius of the two pans are in the ratio 2:1. In other words we can say if the radius of the larger pan is ‘r’ the radius of the smaller pan is r/2 and since they are similar their heights are also in the same ratio 2:1. So, if h is the height of the larger triangle, h/2 is the height of the smaller triangle.

Now,

Volume of the larger cylinder Vl= pi r2h

Volume of the smaller cylinder Vs=pi(r/2)2(h/2) =(pir2h )/8 = Vl/8

i.e Vs =Vl /8

Therefore 8 smaller pans can fill one larger pan.

Black & White

published on February 27th, 2008 . by Vanaja

The following figures represent a relationship between two variables.

linear.JPG

Which rule relates x the number of dark squares to y, the number of white squares?

 

Ans:

y=2x+3

A Sales Chart

published on February 17th, 2008 . by Vanaja

Sarah went to a grocery shop. There was a sales chart.

Items Price
Cabbage $1.8 for 2
Carrot $0.6 for 1.k.g
Onion $1.75 for 1 carton
Potato $2.05 for 1 carton
Tomato $1 for 1 k.


Sarah bought 4 carton potatoes and 1 cabbage. If she gave $10, how much money she got back?



Hint: 10-{(4×2.05)+.9}

A football problem

published on August 21st, 2007 . by Vanaja

At a football championship 600 tickets were sold .
Child ticket cost $2 each and adult ticket cost $5 each. The total money collected for the game was $1650.

Find the number of tickets sold in each category.
Hint:
Let x be number of children and y be number of adults.
x+y = 600
2x+5y= 1650
==>x= 450, y=150

Fundamental Theorem of Arithmetic

published on April 13th, 2007 . by Vanaja

Fundamental Theorem of Arithmetic :

Every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

The Fundamental Theorem of Arithmetic says that every composite number can be factorised as a product of primes. Actually it says more. It says that given any composite number it can be factorised as a product of prime numbers in a‘unique’ way, except for the order in which the primes occur. That is, given any composite number there is one and only one way to write it as a product of primes,as long as we are not particular about the order in which the primes occur.

So, for example, we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written. This fact is also stated in the following form:

The prime factorisation of a natural number is unique, except for the order of its factors.

In general, given a composite number x, we factorise it as x = p1p2 … pn, where p1, p2,…, pn are primes and written in ascending order

If we combine the same primes, we will get powers of primes.

Once we have decided that the order will be ascending, then the way the number is factorised, is unique.

The Fundamental Theorem of Arithmetic has many applications, both within mathematics and in other fields.

Euclid’s Division Lemma

published on April 10th, 2007 . by Vanaja

Euclid’s Division Lemma
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0?r

This result was perhaps known for a long time, but was first recorded in Book VII of Euclid’s Elements. Euclid’s division algorithm is based on this lemma.
Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.
Let us see how the algorithm works, through an example first. Suppose we need to find the HCF of the integers 455 and 42.

We start with the larger integer, that is, 455.
Then we use Euclid’s lemma to get 455 = 42 × 10 + 35.
Now consider the divisor 42 and the remainder 35, and apply the division lemma to get 42 = 35 × 1 + 7.
Now consider the divisor 35 and the remainder 7, and apply the division lemma to get 35 = 7 × 5 + 0.

Notice that the remainder has become zero, and we cannot proceed any further. We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7. You can easily verify this by listing all the factors of 455 and 42.Why does this method work? It works because of the following result. So, let us state Euclid’s division algorithm clearly.To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ?rStep 2 : If r = 0, d is the HCF of c and d. If r ? 0, apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

This algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d, etc.Euclid’s division algorithm is not only useful for calculating the HCF of very large numbers, but also because it is one of the earliest examples of an algorithm that a computer had been programmed to carry out.

Remarks :
1. Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also.
2. Although Euclid’s Division Algorithm is stated for only positive integers, it can be extended for all integers except zero, i.e., b ? 0.

Some Properties Of Ratios

published on December 9th, 2006 . by Vanaja

1) Let a:b>c:d and c:d be two ratios. Then,

i) a:b > c:d, if ad>bc,

ii) a:b< ?xml:namespace prefix = c /> <>

iii) a:b = c:d, if ad=bc

2) A ratio a:b is called a ratio of

i) greater inequality ifa>b,

ii) less inequality if a < b

iii) equality if a=b

3) If the same positive quantity is added to both the terms of a ratio of greater inequality, then the ratio is decreased.

4) If the same positive quantity is added to both the terms of a ratio of less inequality, then the ratio is decreased.

5) If the same positive quantity is subtracted to both the terms of a ratio of greater inequality, then the ratio is increased.

6) If the same positive quantity is added to both the terms of a ratio of greater inequality, then the ratio is increased.

Ratio

published on December 5th, 2006 . by Vanaja

The ratio of two quantities of the same kind and in the same units is a comparison by division of the measure of two quantities.
In other words ,the ratio of two quantities of the same kind is the relation between their measures and determines how many times one quantity is greater than or less than the other quantity.
The ratio of a to b is the fraction a/b, and is generally written as a:b.

  • Example 1: The ratio of $25 to $50 is 25:50 or25/50 or 1:2
  • Example 2: The ratio of 2m to 80 cm is 200:80 or 200/80 or 5:2
  • Example 3: There is no ratio between $10 and 5 meter.

Since the ratio of two quantities of the same kind determines how many times one quantity contains other, is an abstract quantity. In other words, ratio has no unit or it is independent of the units used in the quantities compared.

For the ratio a:b, a and b are called terms of the ratio. The former a is called the first term or antecedent and the later b is known as the second term or consequent.

Periodic Functions

published on November 16th, 2006 . by Vanaja

Periodic functions are functions that repeat its values over and over, after some definite period or cycle on a specific period. This can be expressed mathematically that A function f is said to be periodic if there exists a real T>0 such that f (x+T) = f(x) for all x.

The fundamental period of a function is the length of a smallest continuous portion of the domain over which the function completes a cycle. That is, it’s the smallest length of domain that if you took the function over that length and made an infinite number of copies of it, and laid them end to end, you would have the original function.

If a function is periodic, then the smallest t>0 ,if it exists such that f (x+t) = f(x) for all x, is called the fundamental period of the function.

The trigonometric functions sine and cosine are common periodic functions, with period 2?.
ie. sin (x+2?)= sin x , cos(x+2?)=cos x

But tan and cot remain unchanged when x is increased by pi.
ie. tan(x+?)=tan x, cot(x+?)= cot x
So, they are periodic functions with period ? .

An aperiodic function (non-periodic function) is one that has no such period

Rational Root(Zero) Theorem

published on November 9th, 2006 . by Vanaja

Roots of a Polynomial

A root or zero of a function is a number that, when plugged in for the variable, makes the function equal to zero. Thus, the roots of a polynomial P(x) are values of x such that P(x) = 0.

The Rational Zeros (Roots) Theorem

We can use the Rational Zeros Theorem to find all the rational zeros of a polynomial.

  1. Arrange the polynomial in descending order
  2. Write down all the factors of the constant term. These are all the possible values of p.
  3. Write down all the factors of the leading coefficient (The coefficient of the first term of a polynomialwhen writing in descending order.)
    These are all the possible values of q.
  4. Write down all the possible values of p/q . Remember that since factors can be negative p/q, and - (p/q)must both be included. Simplify each value and cross out any duplicates.
  5. Use synthetic division or remainder theorem to determine the values of p/q for which P(p/q) = 0. These are all the rational roots of P(x).

Example:

Find all the possible rational roots of

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